Question

# What information is contained in the phase spectrum of a signal?

I often read articles in which the authors compute both the phase and the amplitude spectra. I know what the amplitude spectrum is but I just can't make sense of the phase spectrum.

Sudipta Das
National Institute of Technology, Durgapur
If you compare the amplitude spectra of a reference signal with its corrupted version, you get the information about the way noise effected this signal.
Consequently, if you compare the phase spectra of these two signals, you get the delay spread information or dispersion, the information about phase distortion.
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When you perform any operation on a signal (say a filter) that will affect the amplitude and the phase of all frequency components of your signal. You understand what the amplitude change is. The phase simply reflects the (group) delay for each of the frequency components. The (group) delay is -d(phase)/d(w), where w=2*pi*f. An interesting kind of system is that for which the phase is linear. If that is the case then the delay (its derivative) is flat. If your operation manages to do that then the delay of all frequencies you are interested in is the same and there is no shape distortion of your signal (when you add them all up to make up your signal they are delayed by the same amount and so they align perfectly in the same way they were before the filter). This is important for diagnostics (say you are dealing with an electrocardiogram, for instance).
In terms of filters, the family of filters that has the flattest group delay (the most linear phase) is the Bessel filter, while the family that has the narrowest transition band is also the one that has the poorest phase response (less linear), especially around the cut-off frequency - that is the elliptic or Cauer filter.
For a single frequency the phase helps determine causality (which occurred first) or tracking the path of the signal (in EEG signals for example).
Hope this helps shed some light on it.
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Lavdim Kurtaj
University of Prishtina, Faculty of Electrical and Computer Engineering, Kosovo, Prishtina
One of possible views of the signal f(t) is that it is composed of a number of sinusoidal signals (sin(...)) with given frequency (f), amplitude (A) and phase (Phi). Graphical presentation of these two quantities as function of frequency gives amplitude spectrum (characteristics) and phase spectrum (characteristics).
f(t) -> Sum( A*sin(2*pi*f*t+Phi) )
Amplitude and phase, for a given frequency and given signal, can be calculated from Fourier transform. Both of them usually are a function of frequency, i.e. A(f) and Phi(f).
Lets suppose that signal f(t) is periodical. If you pass this signal through a very narrow band-pas filter (BPF(f)) with its center frequency at f, then output would be sinusoidal signal with amplitude A(f) and phase Phi(f), and they correspond to one point in amplitude spectrum and one point of phase spectrum. Phi(f) in radians (or degree) is measured relative to periodicity of input signal.
f(t) -> BPF(f) -> A(f)*sin(2*pi*f*t+Phi(f))
Classical Spectral-Analyzers plotted on screen these two quantities in vertical axis, while horizontal axis and center of BPF were changed linearly (axis position from left-to-right and center of BPF frequency from fmin to fmax). Most modern Spectral-Analyzers calculate A(f) and Phi(f) by Fast-Fourier-Transform (FFT).
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By the common words, the phase spectrum shows the phase shifts between signals with different frequencies. The very simple example is the chromatic dispersion. Assume the signal has definite phase shifts at the input in some volume with dispersive medium. At the output of volume the phase pattern become different, because at the chromatic dispersion the different frequences have different speed in the medium.
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If possible, you refer to the following paper.
H. Kitayoshi,“A New Detection Technique for Digital Radio Communication Systems:Applied Spectrum Phase Interpolation”, 1994 IEEE Supercomm/ICC, NewOrleans,U.S.A., Conference Record vol. 3, pp. 1417-1421, May 1994.
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In radio astronomy the relative phase of signals on two antennnas says something about the angular location and distribution of the radio source. In radar interferometry (c.f. works by Hysell and Chau) one computes the Doppler power spectrum among several antennas, having both both magnitude and phase, which permits one to deduce independently the angular distribution of targets at the same range with different Doppler content.
There is also an interesting relationship between the magnitude and phase spectrum of the dielectric function of matter (the Kramers-Kronig Relations) which results from the requirement that matter behaves in a causal fashion. That is, careful measurements of the amplitude spectrum permit inference about the phase spectrum (and vice versa).
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By observing phase spectrum you could know the information that one can extract from amplitude spectrum only in case of minimum phase system. because phase spectrum of minimum phase system is uniquely related to amplitude spectrum, while this is not true for non minimum phase system. here i am citing some of them that we can extract from phase spectrum :
(1) By observing the slope of phase response we can estimate relative stability. Relative stability is a very useful tool in designing of a system. For an oscillator slope of phase response should be as high as possible. if there is an abrupt change in phase response ( i.e. slope is infinite ) then it is highly stable oscillator in term of frequency.
(2) We could know that parameter like Q factor by phase response and hence damping ratio ( because damping ratio is inversely proportional to Q factor ) . if one know the value of damping ratio then one can easily predict their transient behavior.
(3) In series RLC circuit less steeper phase response has high Q value means they produce clean oscillations at resonance and selectivity is also very high i.e. they pickup only a single frequency out of a band of frequency ( RLC circuit used as band reject filter and used in super heterodyne receiver circuit ) . For parallel RLC circuit this condition gets reversed .( more steeper phase response means high Q factor )
(4) Q factor is also related to power dissipation . in parallel RLC circuit a high Q value is achieved via a large value of R resulting high loss. while In series RLC high Q value means low power dissipation
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When you perform any operation on a signal (say a filter) that will affect the amplitude and the phase of all frequency components of your signal. You understand what the amplitude change is. The phase simply reflects the (group) delay for each of the frequency components. The (group) delay is -d(phase)/d(w), where w=2*pi*f. An interesting kind of system is that for which the phase is linear. If that is the case then the delay (its derivative) is flat. If your operation manages to do that then the delay of all frequencies you are interested in is the same and there is no shape distortion of your signal (when you add them all up to make up your signal they are delayed by the same amount and so they align perfectly in the same way they were before the filter). This is important for diagnostics (say you are dealing with an electrocardiogram, for instance).
In terms of filters, the family of filters that has the flattest group delay (the most linear phase) is the Bessel filter, while the family that has the narrowest transition band is also the one that has the poorest phase response (less linear), especially around the cut-off frequency - that is the elliptic or Cauer filter.
For a single frequency the phase helps determine causality (which occurred first) or tracking the path of the signal (in EEG signals for example).
Hope this helps shed some light on it.
18 Recommendations Amplitude and phase are both basic and independent informations about a signal after a harmonic decomposition. While the amplitude tells you how strong is a harmonic in a signal, the phase tells where this harmonic lies in time. Applied on your hearing, you can estimate the strength of the sirene of an rescue car (amplitude in the acoustic spectrum) and you can estimate the direction the car is coming from based on the phase (difference) of the sound coming into your ears. Simply, the amplitude ist the strength of a signal, the phase ist its time or direction appearance.
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You can make a very simple check, by reproducing a magnetic tape recording in reverse direction. This does not change the amplitude spectrum of audio data, only phase spectrum is affected.
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Peter Husar:
From your answer it can be easily deduced that our ear is sensitive to phase distortion. is it really true ? not necessary to phase distortion, however this is true, but the auditory system is definitely sensitive to phase (difference) information of received sound. Otherwise you could not detect the direction of incoming sound.
Marc Schmieger
Fraunhofer Institute for Biomedical Engineering IBMT
A practical usage for Nandan`s statements is measuring the impedance curve of a piezoelectric sensor over a free chosen frequency range. You also get magnitude |Z| and phase curve phi. So you are able to see how the transducer/sensor behave at his resonance frequency (lowest impedance).
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While the information in the previous answers is interesting and correct, imho they do not really answer the question, except the answer from Peter Husar. The phase of a signal gives information about the relation between the different frequencies. The phase determines where the signal energy will be localized in time. For example, the Dirac impulse function has all of its energy concentrated at a single moment in time, while a white noise signal has all energy spread out evenly in time. Nevertheless, both signals has an identical (flat) amplitude spectrum. The impulse has all phases aligned, whereas the white noise obviously has all phases randomly.
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I give you two different signals with the same amplitude spectrum but with different phase spectra. Are the signals gonna like identical in the time domain. Hell, no. This means that the phase spectrum is an essential piece of info (along with the magnitude). Alternatively, one can represent the signal by its I and Q (in phase and quadrature components). Complex representations of signals always have two components, magnitude and phase or real and imaginary parts.
A related reason why phase is so important is that nature (quantum mechanics, electronmagnetics) is based on INTERFERENCE. Waves of given magnitudes interact via their phases in order to yield the amplitudes of their resultants. For example adding two identical sinusoids of the same phase doubles it, whereas adding two sinusoids of the same amplitudes but phases with 180 deg difference wipes the resultant out.
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Marc Schmieger
Fraunhofer Institute for Biomedical Engineering IBMT
@Peter Husar
Active Noise Cancellation (ANC) e.g. in headphones gives an good impression, how necessary phase information is in auditory systems.
@Moshe Nazarathy
An analogy to your mentioned I and Q signal could be the torsional moment.
A given weight m (e.g. 1 litre beer) holding straight above your head applies a force F down to earth. The length of your arm will be equatable to the distance (radius) r. The torsional moment M will be the smallest, as long as you start moving your arm to the horizontal position (change angle/ phase). M will reach the maximum in horizontal position. The angle of arm position visualizes the importance of phase information although the weight m (amplitude) still remains the same.
:-)
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Sudipta Das
National Institute of Technology, Durgapur
If you compare the amplitude spectra of a reference signal with its corrupted version, you get the information about the way noise effected this signal.
Consequently, if you compare the phase spectra of these two signals, you get the delay spread information or dispersion, the information about phase distortion.
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